Q.
Choose the statement which is incorrect among the following.
I. $\cot x$ decreases from 0 to $-\infty$ in first quadrant and increases from 0 to $\infty$ in third quadrant.
II. $\sec x$ increases from $-\infty$ to $-1$ in second quadrant and decreases from $\infty$ to 1 in fourth quadrant.
III. $\operatorname{cosec} x$ increases from 1 to $\infty$ in second quadrant and decreases from $-1$ to $-\infty$ in fourth quadrant.
Trigonometric Functions
Solution:
From the figure it is clear that in the first quadrant, as $x$ increases from 0 to $\frac{\pi}{2}, \sin x$ increases from 0 to 1 .
In the second quadrant, as $x$ increases from $\frac{\pi}{2}$ to $\pi, \sin x$ decreases from 1 to 0 . In the third quadrant, as $x$ increases from $\pi$ to $\frac{3 \pi}{2}, \sin x$ decreases from 0 to $-1$ and finally, in the fourth quadrant, $\sin x$ increases from $-1$ to 0 as $x$ increases from $\frac{3 \pi}{2}$ to $2 \pi$. Similarly, we get the behaviour of other trigonometric functions and get the following table.
I quadrant
II quadrant
III quadrant
IV quadrant
sin
increases
from 0 to 1
decreases
from 1 to 0
decreases
from 0 to -1
increases
from 0 to 1
cos
decreases
from 1 to 0
decreases
from 0 to -1
increases
from $-1$ to 0
increases
from 0 to 1
tan
increases
from 0 to $\infty$
increases
from $-\infty$ to 0
increases
from 0 to $\infty$
increases
from $-\infty$ to 0
cot
decreases
from $\infty$ to 0
decreases
from 0 to $-\infty$
decreases
from $\infty$ to 0
decreases
from 0 to $-\infty$
sec
increases
from 1 to $\infty$
increases
from $-\infty$ to $-1$
decreases
from -1 to $-\infty$
decreases
from $\infty$ to 1
cosec
decreases
from $\infty$ to 1
increases
from 1 to $\infty$
increases
from $-\infty$ to $-1$
decreases
from -1 to $-\infty$
Remark In the above table, the statement $\tan x$ increases from 0 to $-\infty$ (infinity) for $0 < x < \frac{\pi}{2}$ simply means that $\tan x$ increases as $x$ increases for $0 < x < \frac{\pi}{2}$ and assumes arbitrarily large positive values as $x$ approaches to $\frac{\pi}{2}$. Similarly, to say that $\operatorname{cosec} x$ decreases from $-1$ to $-\infty$ (minus infinity) in the fourth quadrant means that $\operatorname{cosec} x$ decreases for $x \in\left(\frac{3 \pi}{2}, 2 \pi\right)$ and assumes arbitrarily large negative values as $x$ approaches to $2 \pi$. The symbols $\infty$ and $-\infty$ simply specify certain types of behaviour of functions and variables.
Thus, we see that all the given statements are true.
| I quadrant | II quadrant | III quadrant | IV quadrant | |
|---|---|---|---|---|
| sin | increases from 0 to 1 | decreases from 1 to 0 | decreases from 0 to -1 | increases from 0 to 1 |
| cos | decreases from 1 to 0 | decreases from 0 to -1 | increases from $-1$ to 0 | increases from 0 to 1 |
| tan | increases from 0 to $\infty$ | increases from $-\infty$ to 0 | increases from 0 to $\infty$ | increases from $-\infty$ to 0 |
| cot | decreases from $\infty$ to 0 | decreases from 0 to $-\infty$ | decreases from $\infty$ to 0 | decreases from 0 to $-\infty$ |
| sec | increases from 1 to $\infty$ | increases from $-\infty$ to $-1$ | decreases from -1 to $-\infty$ | decreases from $\infty$ to 1 |
| cosec | decreases from $\infty$ to 1 | increases from 1 to $\infty$ | increases from $-\infty$ to $-1$ | decreases from -1 to $-\infty$ |